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==Example==
Consider for example a sequence of [[random variable]]s each of which takes value from the ternary [[alphabet]] ''{a,b,c}''.
Consider for example the string ''aaabcaaabcaaabcaaabc…aaabc'' constructed from infinite concatenations of the sub-string ''aaabc''.
To construct the [[Markov chain]] of order 1 for the next character in this sequence, one needs to estimate the following 9 conditional probability components {Pr(a|a), Pr(a|b), Pr(a|c), Pr(b|a), Pr(b|a), Pr(b|a), Pr(c|a), Pr(c|a), Pr(c|a)}.
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The Variable Order Markov model assumes that in realistic settings, there are certain realizations of states (represented by contexts) in which some past states are independent from the future states, accordingly, a great reduction in the number of model parameters can be achieved.
The VOM model of maximal order 2 can approximate the above string using ''only'' the following four [[conditional probability]] components {Pr(a|aa)=0.5, Pr(b|aa)=0.5, Pr(c|b)=1.0, Pr(a|c)= 1.0}. In this example, Pr(c|ab)=Pr(c|b)=1.0, therefore, the shorter context b is sufficient to determine the future state. Similarly, the VOM model of maximal order 3 can approximate the string using only four [[conditional probability]] components.
==Application Areas==
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